When Interpol seized an estimated $2.6 million worth of drugs in a blitz last fall, the international crime fighters weren’t after cocaine or heroin. They confiscated thousands of packages of antibiotics, insulin and heart and cancer medications.

The sting, dubbed Operation Pangea III, was part of a less familiar war on drugs — the fight against counterfeit meds. It’s a high-stakes battle: A $1,000 investment in fake drugs can return $30,000, 10 times the typical profit from the same invest...